CHM 374 Spring 2010 HOMEWORK SET 7 Due Wed, March 24th Read Understanding Physical Chemistry ( UPC ), Chapter 7, Section 7.1 1. Note that all of the eigenfunctions of the hydrogen atom Hamiltonian, H ˆ , are also eigenfunctions of 2 ˆ L . a. What are the values of E H (in J units) and 2 L (in J s units) for the following hydrogen orbitals? i) 1s ii) 2s iii) 2p iv) 3p v) 3d b. Apply the z L ˆ operator to the 0, 1 , 2 , 1 , 1 , 2 , and 1 , 1 , 2 orbitals of hydrogen to demonstrate that they are each eigenfunctions of z L ˆ with eigenvalues equal to l m . c. Without performing any calculations, explain why only one of the three 2p orbitals of hydrogen is an eigenfunction of z L ˆ , and predict the expectation value z L for each of the three 2p orbitals (again without doing any explicit calculations). Hint: Recall that two of the 2p orbitals are real linear combinations of the complex 1 , 1 , 2 , and 1 , 1 , 2 orbitals, and make use of the theorem that is proven on p. 212.
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